Theorem isorcl2 49663

Description: Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
isorcl.i	⊢ 𝐼 = (Iso‘𝐶)
isorcl.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
isorcl2.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
isorcl2	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Proof of Theorem isorcl2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isorcl.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
2		isorcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2765	. . . . 5 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		isorcl.i	. . . . . 6 ⊢ 𝐼 = (Iso‘𝐶)
5	4, 1	isorcl 49662	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	2, 3, 5, 4	isofval2 49661	. . . 4 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦)))
7	6	oveqd 7417	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌))
8	1, 7	eleqtrd 2867	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌))
9		eqid 2765	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))
10	9	elmpocl 7641	. 2 ⊢ (𝐹 ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
11	8, 10	syl 18	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  dom cdm 5652  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17259  Invcinv 17792  Isociso 17793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-inv 17795  df-iso 17796

This theorem is referenced by:  isoval2  49664  catcisoi  50029